标题： 稳定相位恢复：泊松和重尾模型中的最优率 显示英文标题

标题： Stable Phase Retrieval: Optimal Rates in Poisson and Heavy-tailed Models

摘要： 我们研究了在两种现实且具有挑战性的噪声模型下相位恢复的稳定恢复保证：泊松模型和重尾模型。 我们的分析涵盖了非凸最小二乘（NCVX-LS）和凸最小二乘（CVX-LS）估计量。 对于泊松模型，我们证明了在真实信号$pmb{x}$超过某个能量阈值的高能量范围内，两种估计量都能达到与信号无关的、极小极大最优误差率$\mathcal{O}(\sqrt{\frac{n}{m}})$，其中$n$表示信号维度，$m$表示采样向量的数量。 相反，在低能量范围内，NCVX-LS 估计量达到误差率$\mathcal{O}(\|\pmb{x}\|^{1/4}_2\cdot(\frac{n}{m})^{1/4})$，该误差率随着信号$\pmb{x}$的能量减小而减小，并且相对于过采样比而言几乎是最优的。 这表明在泊松设置中存在与信号能量自适应的行为。 对于具有有限$q$-阶矩（$q>2$）噪声的重尾模型，两个估计量在高能量区域中达到最小最大最优误差率$\mathcal{O}( \frac{\| \xi \|_{L_q}}{\| \pmb{x} \|_2} \cdot \sqrt{\frac{n}{m}} )$，而NCVX-LS估计量在低能量区域中进一步达到最小最大最优率$\mathcal{O}( \sqrt{\|\xi \|_{L_q}}\cdot (\frac{n}{m})^{1/4} )$。我们的分析基于两个关键思想：利用乘子不等式处理可能依赖于采样向量的噪声，以及对泊松噪声的新解释，在高能量区域中将其视为次指数，在低能量区域中则视为重尾。这些见解构成了统一分析框架的基础，我们进一步将其应用于一系列相关问题，包括稀疏相位检索、低秩PSD矩阵恢复和随机盲卷积。 显示英文摘要

摘要： We investigate stable recovery guarantees for phase retrieval under two realistic and challenging noise models: the Poisson model and the heavy-tailed model. Our analysis covers both nonconvex least squares (NCVX-LS) and convex least squares (CVX-LS) estimators. For the Poisson model, we demonstrate that in the high-energy regime where the true signal $pmb{x}$ exceeds a certain energy threshold, both estimators achieve a signal-independent, minimax optimal error rate $\mathcal{O}(\sqrt{\frac{n}{m}})$, with $n$ denoting the signal dimension and $m$ the number of sampling vectors. In contrast, in the low-energy regime, the NCVX-LS estimator attains an error rate of $\mathcal{O}(\|\pmb{x}\|^{1/4}_2\cdot(\frac{n}{m})^{1/4})$, which decreases as the energy of signal $\pmb{x}$ diminishes and remains nearly optimal with respect to the oversampling ratio. This demonstrates a signal-energy-adaptive behavior in the Poisson setting. For the heavy-tailed model with noise having a finite $q$-th moment ($q>2$), both estimators attain the minimax optimal error rate $\mathcal{O}( \frac{\| \xi \|_{L_q}}{\| \pmb{x} \|_2} \cdot \sqrt{\frac{n}{m}} )$ in the high-energy regime, while the NCVX-LS estimator further achieves the minimax optimal rate $\mathcal{O}( \sqrt{\|\xi \|_{L_q}}\cdot (\frac{n}{m})^{1/4} )$ in the low-energy regime. Our analysis builds on two key ideas: the use of multiplier inequalities to handle noise that may exhibit dependence on the sampling vectors, and a novel interpretation of Poisson noise as sub-exponential in the high-energy regime yet heavy-tailed in the low-energy regime. These insights form the foundation of a unified analytical framework, which we further apply to a range of related problems, including sparse phase retrieval, low-rank PSD matrix recovery, and random blind deconvolution.